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| In [[topology]], the '''wedge sum''' is a "one-point union" of a family of [[topological space]]s. Specifically, if ''X'' and ''Y'' are [[pointed space]]s (i.e. topological spaces with distinguished basepoints ''x''<sub>0</sub> and ''y''<sub>0</sub>) the wedge sum of ''X'' and ''Y'' is the [[quotient space]] of the [[disjoint union (topology)|disjoint union]] of ''X'' and ''Y'' by the identification ''x''<sub>0</sub> ∼ ''y''<sub>0</sub>:
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| :<math>X\vee Y = (X\amalg Y)\;/ \sim,\,</math>
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| where ∼ is the [[equivalence closure]] of the relation {(''x''<sub>0</sub>,''y''<sub>0</sub>)}.
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| More generally, suppose (''X''<sub>''i''</sub>{{pad|0.1em}})<sub>''i''∈''I''</sub> is a [[indexed family|family]] of pointed spaces with basepoints {''p''<sub>''i''</sub>{{pad|0.1em}}}. The wedge sum of the family is given by:
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| :<math>\bigvee_i X_i = \coprod_i X_i\;/ \sim,\,</math>
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| where ∼ is the equivalence relation {(''p<sub>i</sub>''{{pad|0.1em}}, ''p<sub>j</sub>''{{pad|0.1em}}) | ''i,j'' ∈ ''I''{{pad|0.1em}}}.
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| In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints {''p''<sub>''i''</sub>}, unless the spaces {''X''<sub>''i''</sub>{{pad|0.1em}}} are [[homogeneous space|homogeneous]].
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| The wedge sum is again a pointed space, and the binary operation is [[associative]] and [[commutative]] ([[up to isomorphism]]).
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| Sometimes the wedge sum is called the '''wedge product''', but this is not the same concept as the [[exterior product]], which is also often called the wedge product.
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| ==Examples== | |
| The wedge sum of two circles is [[homeomorphic]] to a [[figure-eight space]]. The wedge sum of ''n'' circles is often called a ''[[bouquet of circles]]'', while a wedge product of arbitrary spheres is often called a '''bouquet of spheres'''.
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| A common construction in [[homotopy]] is to identify all of the points along the equator of an ''n''-sphere <math>S^n</math>. Doing so results in two copies of the sphere, joined at the point that was the equator:
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| :<math>S^n/{\sim} = S^n \vee S^n </math>
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| Let <math>\Psi</math> be the map <math>\Psi:S^n\to S^n \vee S^n</math>, that is, of identifying the equator down to a single point. Then addition of two elements <math>f,g\in\pi_n(X,x_0)</math> of the ''n''-dimensional [[homotopy group]] <math>\pi_n(X,x_0)</math> of a space ''X'' at the distinguished point <math>x_0\in X</math> can be understood as the composition of <math>f</math> and <math>g</math> with <math>\Psi</math>:
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| :<math>f+g = (f \vee g) \circ \Psi</math>
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| Here, <math>f</math> and <math>g</math> are understood to be maps, <math>f:S^n\to X</math> and similarly for <math>g</math>, which take a distinguished point <math>s_0\in S^n</math> to a point <math>x_0\in X</math>. Note that the above defined the wedge sum of two functions, which was possible because <math>f(s_0)=g(s_0)=x_0</math>, which was the point that is equivalenced in the wedge sum of the underlying spaces.
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| ==Categorical description== | |
| The wedge sum can be understood as the [[coproduct]] in the [[category of pointed spaces]]. Alternatively, the wedge sum can be seen as the [[pushout (category theory)|pushout]] of the diagram ''X'' ← {•} → ''Y'' in the [[category of topological spaces]] (where {•} is any one point space).
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| ==Properties==
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| [[Van Kampen's theorem]] gives certain conditions (which are usually fulfilled for [[well-behaved]] spaces, such as [[CW complex]]es) under which the [[fundamental group]] of the wedge sum of two spaces ''X'' and ''Y'' is the [[free product]] of the fundamental groups of ''X'' and ''Y''.
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| ==See also==
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| *[[Smash product]]
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| *[[Hawaiian earring]], a topological space resembling, but not the same as, a wedge sum of countably many circles
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| ==References==
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| * Rotman, Joseph. ''An Introduction to Algebraic Topology'', Springer, 2004, p. 153. ISBN 0-387-96678-1
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| {{DEFAULTSORT:Wedge Sum}}
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| [[Category:Topology]]
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| [[Category:Binary operations]]
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| [[Category:Homotopy theory]]
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